37 research outputs found
Unifying Projected Entangled Pair States contractions
The approximate contraction of a Projected Entangled Pair States (PEPS)
tensor network is a fundamental ingredient of any PEPS algorithm, required for
the optimization of the tensors in ground state search or time evolution, as
well as for the evaluation of expectation values. An exact contraction is in
general impossible, and the choice of the approximating procedure determines
the efficiency and accuracy of the algorithm. We analyze different previous
proposals for this approximation, and show that they can be understood via the
form of their environment, i.e. the operator that results from contracting part
of the network. This provides physical insight into the limitation of various
approaches, and allows us to introduce a new strategy, based on the idea of
clusters, that unifies previous methods. The resulting contraction algorithm
interpolates naturally between the cheapest and most imprecise and the most
costly and most precise method. We benchmark the different algorithms with
finite PEPS, and show how the cluster strategy can be used for both the tensor
optimization and the calculation of expectation values. Additionally, we
discuss its applicability to the parallelization of PEPS and to infinite
systems (iPEPS).Comment: 28 pages, 15 figures, accepted versio
Algorithms for finite Projected Entangled Pair States
Projected Entangled Pair States (PEPS) are a promising ansatz for the study
of strongly correlated quantum many-body systems in two dimensions. But due to
their high computational cost, developing and improving PEPS algorithms is
necessary to make the ansatz widely usable in practice. Here we analyze several
algorithmic aspects of the method. On the one hand, we quantify the connection
between the correlation length of the PEPS and the accuracy of its approximate
contraction, and discuss how purifications can be used in the latter. On the
other, we present algorithmic improvements for the update of the tensor that
introduce drastic gains in the numerical conditioning and the efficiency of the
algorithms. Finally, the state-of-the-art general PEPS code is benchmarked with
the Heisenberg and quantum Ising models on lattices of up to
sites.Comment: 18 pages, 20 figures, accepted versio
Adiabatic Preparation of a Heisenberg Antiferromagnet Using an Optical Superlattice
We analyze the possibility to prepare a Heisenberg antiferromagnet with cold
fermions in optical lattices, starting from a band insulator and adiabatically
changing the lattice potential. The numerical simulation of the dynamics in 1D
allows us to identify the conditions for success, and to study the influence
that the presence of holes in the initial state may have on the protocol. We
also extend our results to two-dimensional systems.Comment: 5 pages, 4 figures + Supplementary Material (5 pages, 6 figures),
published versio
Multigrid Renormalization
We combine the multigrid (MG) method with state-of-the-art concepts from the
variational formulation of the numerical renormalization group. The resulting
MG renormalization (MGR) method is a natural generalization of the MG method
for solving partial differential equations. When the solution on a grid of
points is sought, our MGR method has a computational cost scaling as
, as opposed to for the best standard MG
method. Therefore MGR can exponentially speed up standard MG computations. To
illustrate our method, we develop a novel algorithm for the ground state
computation of the nonlinear Schr\"{o}dinger equation. Our algorithm acts
variationally on tensor products and updates the tensors one after another by
solving a local nonlinear optimization problem. We compare several different
methods for the nonlinear tensor update and find that the Newton method is the
most efficient as well as precise. The combination of MGR with our nonlinear
ground state algorithm produces accurate results for the nonlinear
Schr\"{o}dinger equation on grid points in three spatial
dimensions.Comment: 18 pages, 17 figures, accepted versio
Barren plateaus in quantum tensor network optimization
We analyze the barren plateau phenomenon in the variational optimization of
quantum circuits inspired by matrix product states (qMPS), tree tensor networks
(qTTN), and the multiscale entanglement renormalization ansatz (qMERA). We
consider as the cost function the expectation value of a Hamiltonian that is a
sum of local terms. For randomly chosen variational parameters we show that the
variance of the cost function gradient decreases exponentially with the
distance of a Hamiltonian term from the canonical centre in the quantum tensor
network. Therefore, as a function of qubit count, for qMPS most gradient
variances decrease exponentially and for qTTN as well as qMERA they decrease
polynomially. We also show that the calculation of these gradients is
exponentially more efficient on a classical computer than on a quantum
computer.Comment: 26 pages, 7 figure
Bosonic Fractional Quantum Hall States on a Finite Cylinder
We investigate the ground state properties of a bosonic Harper-Hofstadter
model with local interactions on a finite cylindrical lattice with filling
fraction . We find that our system supports topologically ordered
states by calculating the topological entanglement entropy, and its value is in
good agreement with the theoretical value for the Laughlin state. By
exploring the behaviour of the density profiles, edge currents and
single-particle correlation functions, we find that the ground state on the
cylinder shows all signatures of a fractional quantum Hall state even for large
values of the magnetic flux density. Furthermore, we determine the dependence
of the correlation functions and edge currents on the interaction strength. We
find that depending on the magnetic flux density, the transition towards
Laughlin-like behaviour can be either smooth or happens abruptly for some
critical interaction strength.Comment: 9 pages, 8 figure
Tensor network states in time-bin quantum optics
The current shift in the quantum optics community towards large-size
experiments -- with many modes and photons -- necessitates new classical
simulation techniques that go beyond the usual phase space formulation of
quantum mechanics. To address this pressing demand we formulate linear quantum
optics in the language of tensor network states. As a toy model, we extensively
analyze the quantum and classical correlations of time-bin interference in a
single fiber loop. We then generalize our results to more complex time-bin
quantum setups and identify different classes of architectures for
high-complexity and low-overhead boson sampling experiments